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Towards U(N |M) knot invariantfrom ABJM theory
Taro Kimura
Institut de Physique Theorique, CEA Saclay
Mathematical Physics Laboratory, RIKEN
Based on a collaboration with B. Eynard [arXiv:1408.0010]
Taro Kimura (CEA Saclay/RIKEN) Oct. 2014 @ Bonn 1 / 28
Knot theory
QFT interpretation: Wilson loop in Chern–Simons theory[Witten]
SCS =k
4π
∫S3
Tr
(A ∧ dA+
2
3A ∧A ∧A
)
ex.) Jones polynomial: SU(2) CS w/ the fund rep loop
J(K; q) =⟨W�(K)
⟩/⟨W�( )
⟩with q = exp
(2πi
k +N
)Wilson loop: WR(K) = TrR exp
(∮K
A
)
Taro Kimura (CEA Saclay/RIKEN) Oct. 2014 @ Bonn 2 / 28
Generalizations:
HOMFLY polynomial: SU(2) → SU(N)
Colored polynomial: Tr� U → TrR U
We can generalize it in this way, but...
The expression becomes much more complicataed
What’s the systematic dependence on the rank/rep?
Taro Kimura (CEA Saclay/RIKEN) Oct. 2014 @ Bonn 3 / 28
Matrix integral representation [Marino]
ZCS(S3; q) =1
N !
∫ N∏i=1
dxi2π
e− 1
2gsx2i
N∏i<j
(2 sinh
xi − xj2
)2
Wilson loop operator (especially for unknot)
WR( ) → TrR
ex1
. . .
exN
= sλ(R)(ex1 , · · · , exN )
Wilson loop vev ⟨WR( )
⟩CS
=⟨sλ(ex)
⟩matrix
Taro Kimura (CEA Saclay/RIKEN) Oct. 2014 @ Bonn 4 / 28
Another CS-like matrix model:
ABJM matrix model
ZABJM(S3; q) =1
N !2
∫ N∏i=1
dxi2π
dyi2π
e− 1
2gs(x2i−y2i ) (∆N,N (x; y))2
with ∆N,N (x; y) =
∏Ni<j
(2 sinh
xi−xj2
)(2 sinh
yi−yj2
)∏Ni,j
(2 cosh
xi−yj2
)[Kapustin–Willett–Yaakov] [Drukker–Trancanelli] [Marino–Putrov]
A “supersymmetric” CS matrix model w/ U(N |N) sym
cf. U(N |N) supermatrix model:
ZU(N |N) =
∫ N∏i=1
dxi2π
dyi2π
e− 1
gs(W (xi)−W (yi))
∏Ni<j(xi − xj)2(yi − yj)2∏N
i,j(xi − yj)2Taro Kimura (CEA Saclay/RIKEN) Oct. 2014 @ Bonn 5 / 28
Unknot Wilson loop vev with ABJM⟨WR( )
⟩ABJM
=⟨sλ(ex; ey)
⟩Matrix
Our goal
To construct the U(N |N) knot invariant through ABJM
Taro Kimura (CEA Saclay/RIKEN) Oct. 2014 @ Bonn 6 / 28
Contents
1 Introduction
2 U(N |N) Wilson loop
3 Torus knot
4 Topological string
5 Summary
Taro Kimura (CEA Saclay/RIKEN) Oct. 2014 @ Bonn 7 / 28
U(N) unknot invariant:
⟨WR( )
⟩U(N)
=
N∏i<j
q12(λi−λj−i+j) − q− 1
2(λi−λj−i+j)
q12(−i+j) − q− 1
2(−i+j)
= dimq R
Representation R → Young diagram λ = (λ1, · · · , λN )
An analogous expression for U(N |N) vev?
Taro Kimura (CEA Saclay/RIKEN) Oct. 2014 @ Bonn 8 / 28
Unknot Wilson loop vev with ABJM⟨WR( )
⟩ABJM
=⟨sλ(ex; ey)
⟩Matrix
Supersymmetric Schur function −→ associated w/ U(N |N)
sλ(ex; ey) = StrR
(U(x)
−U(y)
)with U(x) = diag(ex1 , · · · , exN )
Taro Kimura (CEA Saclay/RIKEN) Oct. 2014 @ Bonn 9 / 28
A representation for U(N |N):
λ =
N ×Nµ
νt
α1α2
α3α4α5
β1β2
β3β4β5
Frobenius coordinate:
λ = (α1, · · · , αd(λ)|β1, · · · , βd(λ))d(λ) = #diagonal blocks
if there is a box here, sλ(ex; ey) = 0
d(λ) ≤ N
Let’s focus on d(λ) = N :
Determinantal formula for U(N |N) Wilson loop
⟨WR( )
⟩U(N |N)
= det1≤i,j≤N
(1
q12(αi+βj+1) + q−
12(αi+βj+1)
)
Taro Kimura (CEA Saclay/RIKEN) Oct. 2014 @ Bonn 10 / 28
⟨WR( )
⟩U(N |N)
= det1≤i,j≤N
(1
q12(αi+βj+1) + q−
12(αi+βj+1)
)
=
∏Ni<j
(q
12(αi−αj) − q− 1
2(αi−αj)
)(q
12(βi−βj) − q− 1
2(βi−βj)
)∏Ni,j
(q
12(αi+βj+1) + q−
12(αi+βj+1)
)⟨Wµ( )
⟩U(N)
⟨Wν( )
⟩U(N)
Including two U(N) invariants
⟨WR( )
⟩U(N |N)
∼⟨Wµ( )
⟩U(N)
×⟨Wν( )
⟩U(N)
Taro Kimura (CEA Saclay/RIKEN) Oct. 2014 @ Bonn 11 / 28
U(1|1) theory:
λ =
α
β
hook rep: λ = (α|β)
⟨W(α|β)( )
⟩U(1|1)
=1
q12(α+β+1) + q−
12(α+β+1)
Factorization property⟨WR( )
⟩U(N |N)
= det1≤i,j≤N
⟨W(αi|βj)( )
⟩U(1|1)
cf. Giambelli compatibility [Borodin et al.]
Taro Kimura (CEA Saclay/RIKEN) Oct. 2014 @ Bonn 12 / 28
Summary
ABJM as the U(N |N) Chern–Simons matrix model
Wilson loop applied to knot theory
Determinantal formula for the U(N |N) Wilson loop:
⟨WR( )
⟩U(N |N)
= det1≤i,j≤N
(1
q12(αi+βj+1) + q−
12(αi+βj+1)
)∼⟨Wµ( )
⟩U(N)
×⟨Wν( )
⟩U(N)
U(1|1) vev as a building block
Taro Kimura (CEA Saclay/RIKEN) Oct. 2014 @ Bonn 13 / 28
Contents
1 Introduction
2 U(N |N) Wilson loop
3 Torus knot
4 Topological string
5 Summary
Taro Kimura (CEA Saclay/RIKEN) Oct. 2014 @ Bonn 14 / 28
The original matrix model only describes the unknot loop..
(P,Q) torus knot CS matrix model
Z(P,Q)CS =
∫ N∏i=1
dxi2π
e− 1
2gsx2i
N∏i<j
(2 sinh
xi − xj2P
2 sinhxi − xj
2Q
)
[Lawrence–Rozansky] [Beasley] [Kallen]
(P,Q) torus knotP
Q
(P,Q) = (3, 2)
Taro Kimura (CEA Saclay/RIKEN) Oct. 2014 @ Bonn 15 / 28
Torus knot: Adams operation
Adams operation:
⟨WR(KP,Q)
⟩=∑V
cVR,Q
⟨WV (K1,f )
⟩with f =
P
Q
A linear combination of theP
Q-framed unknot vevs
Schur function decomposition: sλ(xQ) =∑ν
cνλ,Qsν(x)
Taro Kimura (CEA Saclay/RIKEN) Oct. 2014 @ Bonn 16 / 28
Torus knot: Spectral curve
Saddle point analysis provides the spectral curve
ex.) (P,Q) = (2, 3)
PQ cuts & (P +Q) sheets
This is just given by SL(2,Z) transform of the unknot curve
[Brini–Eynard–Marino]
What happens for the U(N |N) theory?
Taro Kimura (CEA Saclay/RIKEN) Oct. 2014 @ Bonn 17 / 28
(P,Q) torus knot U(N |N) CS matrix model
Z(P,Q)ABJM =
∫ N∏i=1
dxi2π
dyi2π
e− 1
2gs(x2i−y2i ) ∆N,N
( xP
;y
P
)∆N,N
( xQ
;y
Q
)
Remark:Perturbatively equivalent to CS theory on the squashedL(2, 1) with b2 = P/Q through the analytic continuation
[Hama–Hosomichi–Lee] [Tanaka] [Imamura–Yokoyama]
We can show: Adams operation & spectral curve
Taro Kimura (CEA Saclay/RIKEN) Oct. 2014 @ Bonn 18 / 28
Adams operation
Schur function decomposition:
sλ(xQ) =∑ν
cνλ,Qsν(x) −→ sλ(xQ, yQ) =∑ν
cνλ,Qsν(x, y)
The rep theory of U(N) & U(N |N) in a parallel way
Adams operation for U(N |N) theory⟨WR(KP,Q)
⟩U(N |N)
=∑V
cVR,Q
⟨WV (K1,f )
⟩U(N |N)
Taro Kimura (CEA Saclay/RIKEN) Oct. 2014 @ Bonn 19 / 28
Spectral curveSolving the saddle point equation... in a complicated form
ex.) (P,Q) = (2, 3)
F3F4
F 03
F 04
F0
F3
F4
F 03
F 04
F1
F3
F4
F 03
F 04
F2
F0
F1
F2
F 00
F 01
F 02
F0
F1
F2
F 00
F 01
F 02
F3 F4
F 00
F 04
F 03
F3
F4
F 01
F 04
F 03
F3 F4
F 02
F 04
F 03
F3
F4
F 02F 0
0
F 01
F0
F1 F2 F 02 F 0
0
F 01
F0
F1F2
F 03 F 0
4
Table 1: The cuts of the functions Fk(u) and FQ+l(u) for (P, Q) = (2, 3), where Fk = Fk(u),
F 0k = Fk(u!
12(P�Q)) and F 0
Q+l = FQ+l(u!12(Q�P )). The solid and dotted lines correspond
to the cuts from the first resolvent W (1)(u) and the second resolvent W (2)(u). For example,
we can see F0 = F3, F4 under crossing the corresponding cut of W (1)(u), and F0 = F 03, F 0
4
through the cut from W (2)(u).
multiple angles of 2⇡/(PQ). The total number of the cuts is thus 2PQ. Due to the saddle
point equations they satisfy
Fk(u � i0) = FQ+l(u + i0) for W (1)(u) ,
Fk(u � i0) = FQ+l((u + i0)!12(Q�P )) for W (2)(u) ,
Fk((u � i0)!12(P�Q)) = FQ+l((u + i0)!
12(Q�P )) for W (1)(u) ,
Fk((u � i0)!12(P�Q)) = FQ+l(u + i0) for W (2)(u) .
(5.37)
This means that Fk(u � i0) = FQ+l(u + i0) under crossing the cut from the first resolvent
W (1)(u), Fk(u � i0) = FQ+l((u + i0)!12(Q�P )) for the cut from the second W (2)(u), and so
on. See Table 1 for the case with (P, Q) = (2, 3).
Using these functions we define a function
S(u, f) = S1(u, f) S2(u, f) , (5.38)
19
SL(2,Z) transform of the unknot curve
Taro Kimura (CEA Saclay/RIKEN) Oct. 2014 @ Bonn 20 / 28
Remark 1: “Mirror” expression [Kapustin–Willett–Yaakov]
ZABJM =1
N !(2π)k
∫dNz
(2π)N
N∏i<j
tanh
(zi − zk
2k
) N∏i=1
(2 cosh
zi2
)−1N = 4 SYM with a fundamental & adjoint matter at k = 1
Trivial (P,Q) dependence: The mirror is the same
Remark 2: Torus knot in the lens space L(2, 1) via SL(2,Z)[Jockers–Klemm–Soroush] [Stevan]
Taro Kimura (CEA Saclay/RIKEN) Oct. 2014 @ Bonn 21 / 28
Summary
Another matrix model describing torus knots:
SL(2,Z) transform of the unknot model
Basic properties hold for U(N |N) theory
Adams operation
Spectral curve
Taro Kimura (CEA Saclay/RIKEN) Oct. 2014 @ Bonn 22 / 28
Contents
1 Introduction
2 U(N |N) Wilson loop
3 Torus knot
4 Topological string
5 Summary
Taro Kimura (CEA Saclay/RIKEN) Oct. 2014 @ Bonn 23 / 28
Knot in Topological string
Brane insertion: [Ooguri–Vafa]
Z(K;x)
=⟨
det(1⊗ 1− U ⊗ e−x
) ⟩CS
WKB expansion: Z(K;x) ∼ exp
(1
gs
∫ x
p(x)dx
)with p(x) = lim
gs→0
∞∑n=0
gs
⟨TrUn
⟩CSe−nx
Taro Kimura (CEA Saclay/RIKEN) Oct. 2014 @ Bonn 24 / 28
Topological string for ABJM:
local P1 × P1 =
Brane partition function:
Z(K;x, y) =⟨
Sdet
(1⊗ 1− U ⊗
(e−x
e−y
))⟩ABJM
∼ exp
(1
gs
∫ x
yp(x)dx
)Another definition based on topological recursion/B-model
Taro Kimura (CEA Saclay/RIKEN) Oct. 2014 @ Bonn 25 / 28
Contents
1 Introduction
2 U(N |N) Wilson loop
3 Torus knot
4 Topological string
5 Summary
Taro Kimura (CEA Saclay/RIKEN) Oct. 2014 @ Bonn 26 / 28
Summary
U(N |N) Wilson loop vev from ABJM matrix model
det formula:⟨WR
⟩U(N |N)
= det1≤i,j≤N
⟨W(αi|βj)
⟩U(1|1)
Torus knot U(N |N) matrix model
SL(2,Z) transform of the unknot: Adams op, spectral curve
Topological string
Ooguri–Vafa construction: Brane pair-creation
Taro Kimura (CEA Saclay/RIKEN) Oct. 2014 @ Bonn 27 / 28
Discussion
Is it really a knot invariant?
If so, another definition required:
Skein relation
CFT on the boundary
Topological recursion
Volume conjecture, AJ conjecture, knot homology, etc.
Taro Kimura (CEA Saclay/RIKEN) Oct. 2014 @ Bonn 28 / 28